Question

If A = `[(0, -1, 2),(4, 3, -4)]` and B = `[(4, 0),(1, 3),(2, 6)]`, then verify that: (A′)′ = (AB)' = B'A'

Answer 1

Given that: A = `[(0, -1, 2),(4, 3, -4)]`, B = `[(4, 0),(1, 3),(2, 6)]`

L.H.S. AB = `[(0,-1, 2),(4, 3, -4)]_(2 xx 3) [(4, 0),(1, 3),(2, 6)]_(3 xx 2)`

= `[(0 - 1+ 4, 0 - 3 + 12),(16 + 3 - 8, 0 + 9 - 24)]_(2 xx 2)`

= `[(3, 9),(11, -15)]_(2 xx 2)`

(AB)' = `[(3, 11),(9, -15)]_(2 xx 2)`

R.H.S. B' = `[(4, 0),(1, 3),(2, 6)]^'`

= `[(4, 1, 2),(0, 3, 6)]`

A' = `[(0, -1, 2),(4, 3, -4)]^'`

= `[(0, 4),(-1, 3),(2, -4)]`

B'A' = `[(4, 1, 2),(0, 3, 6)]_(2 xx 3)  [(0, 4),(-1, 3),(2, -4)]_(3 xx 2)`

= `[(0 - 1 + 4, 16 + 3 - 8),(0 - 3 + 12, 0 + 9 - 24)]_(2 xx 2)`

= `[(3, 11),(9, -15)]_(2 xx 2)`

L.H.S. = R.H.S.

Hence, (AB)' = B'A' is verified.